Wisdom of the Crowd: Incorporating Social
Influence in Recommendation Models
Shang Shang∗ , Pan Hui† , Sanjeev R. Kulkarni∗ , and Paul W. Cuff∗
∗ Department
of Electrical Engineering, Princeton University, Princeton NJ, 08540, U.S.A.
Telekom Laboratories, Ernst-Reuter-Platz 7, 10587 Berlin, Germany
∗ {sshang, kulkarni, cuff}@princeton.edu, † [email protected]
† Deutsche
Abstract—Recommendation systems have received considerable attention recently. However, most research has been focused
on improving the performance of collaborative filtering (CF)
techniques. Social networks, indispensably, provide us extra
information on people’s preferences, and should be considered
and deployed to improve the quality of recommendations.
In this paper, we propose two recommendation models, for
individuals and for groups respectively, based on social contagion
and social influence network theory. In the recommendation model
for individuals, we improve the result of collaborative filtering
prediction with social contagion outcome, which simulates the
result of information cascade in the decision-making process.
In the recommendation model for groups, we apply social
influence network theory to take interpersonal influence into
account to form a settled pattern of disagreement, and then
aggregate opinions of group members. By introducing the concept
of susceptibility and interpersonal influence, the settled rating
results are flexible, and inclined to members whose ratings are
“essential”.
Index Terms—recommendation model, social influence, collaborative filtering
I. I NTRODUCTION
With the rise of e-commerce, recommendation systems have
been studied intensively in the context of collaborative filtering
(CF) techniques [1]. Early generations of recommendation
system have already been commercialized and have achieved
great success. Recommendation systems serve as an important
component of online retail and Video on Demand (VoD) such
as Amazon1 and Netflix2 [5]. Recommenders give customized
recommendations to online users on books, movies, and commodities according to their previous preference data.
In order to improve the quality of recommendations, work
has been done mostly based on improving collaborative filtering techniques. Most recommendation systems based on collaborative filtering assume that users are independent, which
ignores the role of social influence in people’s buying decisions. Problems such as “data sparsity”, “cold start”, “shilling
attack” still challenge the design of recommendation systems
[1][3]. Social networks and social influence, on the other side,
can provide us extra information on users’ preferences, but
have received less attention. This may be partially due to the
unavailability of large-scale datasets to analyze in the past.
Recently, the emergence of Online Social Networks (OSN)
1 www.amazon.com
2 www.netflix.com
provides us an opportunity to reconsider the structure and
effects of social networks so as to improve recommendation
results [3][7][18]. He et al. proposed a social network-based
recommendation system (SNRS) in [3], which is a probabilistic model for personalized recommendations. SNRS is
based on homophily among friends but did not take social
influence into consideration and did not provide a solution
for recommendation to groups. Traditionally, recommendation
systems are designed to recommend items to individual users.
However, there are some buying-decisions/activities done by
a group of users such as going to a restaurant or watching
a movie where group recommendations are desired. Surprisingly, recommendation to groups is a nontrivial extension
of recommendation to individuals. Unlike individual’s buying
decisions which are a personal choice (the user may be
influenced by others’ opinions but there will be no compromise
in the final decision), preference of a group reflects not only
each individual’s subjective taste but also the knowledge of
other group members’ opinions. Most of the current group recommendation systems such as POLYLENS [15] and G.A.I.N.
[16] focus on different ways to aggregate preference. However,
the problem of how to make recommendations for a group
with the consideration of interpersonal influence among group
members rather than viewing each individual’s preference
separately is often ignored.
In this paper, we propose two mathematical models of
recommendation systems, for individuals and for groups respectively, based on social contagion and social influence
network theory. In the past few years, study on social contagion finds applications in viral marketing, which uses preexisting social networks to promote new products or brands
to market, without the knowledge of users’ preference records
[20]. Given the information of users’ social networks and
preference data, we use a modified linear threshold contagion
model to simulate the social influence by word of mouth,
and add this effect to collaborative filtering results in order
to provide more effective recommendations to individuals. In
the model of recommendations to groups, in order to consider
people’s “compromised disagreement” among group members,
we introduce social influence under sufficient interpersonal
communication to give recommendations that reflect more than
personal taste.
In a typical setting, there is a list of m users U =
{u1 , u2 , ..., um } and a list of n items I = {i1 , i2 , ..., in }. Each
user uj has a list of items Iuj , which the user has rated or from
which the user’s preferences can be inferred. The ratings can
either be explicit, for example, on a 1-5 scale as in Netflix, or
implicit such as purchases or clicks. These data form a m × n
rating matrix. G = (U, E) is a social network, represented by
an undirected graph, where U is a set of nodes and E is a set
of edges. For all u, v ∈ U, (u, v) ∈ E if u, v are “friends”, in
which case v ∈ N (u) and u ∈ N (v), where N (u) is the set
of neighbors of u. We want to make recommendations for a
target user or a group of users given the above information.
The rest of this paper is organized as follows. We discuss
related work in Section II. We propose our social influential
recommendation models for individuals and for groups in
Section III and Section IV, followed by conclusions and future
work in Section V.
II. R ELATED W ORK
A few recent papers used information of users’ social
networks as a component in their recommendation models.
The work in [4] used friendship matrix to modify the probabilistic matrix factorization proposed in [5]. In [6], Debnath
et al. constructed a social network graph with items as nodes,
which represents human judgement of similarity between
items aggregated over a large population of users to estimate
feature weights for the content-based recommendations. The
work in [7] studied how collaboration should be done in
a recommendation system based on a network of “personal
agents” . The most closely related work is found in [18] and
[3]. The work in [18] proposed to reduce computational cost
by limiting similar users to target user’s immediate friends.
The work in [3] considered the effects of homophily among
friends in a recommendation system and built a probabilistic
model to describe this homophily effect. In [3], the author
assumed that item attributes, user attributes and ratings of
immediate friends are independent of each other and that for
each pair of immediate friends, their ratings on the same item
are identical but with an error term following the distribution
of the histogram of previous rating differences. There are,
to the best knowledge of the authors’, no previous work
considering social influence, which plays an important role in
people’s buying decisions. We use a social contagion model to
add the effects of information cascade to the target user in the
buying decision, in order to improve the result of collaborative
filtering prediction.
In addition to recommendation for individuals, there are
some circumstances that recommendation for a group of
users are desired. There are mainly three categories of recommendation systems for groups [14]: (1) merging sets of
recommendations, e.g. taking intersection among all members’
top-k preference; (2) aggregation of individuals’ ratings for
particular items; (3) construction of group preference models.
POLYLENS [15], a system to recommend movies to a group
of users, merged the results of individual recommendations.
While Yu et al. [17] introduced a group preference model
which merged individual profiles, using a minimization of total
Dalal’s distance, and took this virtual profile as a target user
for recommendations. In this paper, we utilize social influence
network theory [8] as a tool to model the opinion formation
of a group of users under interpersonal influences within the
group.
III. R ECOMMENDATIONS FOR INDIVIDUALS
Traditional collaborative filtering systems suppose that users
are all independent. However, in reality, social influence plays
a crucial role in people’s buying decisions. For example,
a user, Jack, is browsing on Amazon to choose a leisure
book to read, with nothing particular in mind. In the list of
new collections, Jack vaguely remembers that his friend Lisa
mentioned book A in the list. However, it is a romance book,
which is not his type. His eyes then are caught by two other
books B and C. B, a thriller, was strongly recommended by
his friends Henry and James, and C, though, he has never
heard of it, also looks interesting. Finally, Jack puts C in wish
list and decides to buy book B first. If we look closely at
this scenario, and think about how we make our decisions
everyday, we can see that what drives us to buying decisions
are not only our own preferences but also some informative
comments made by friends. In addition, studies show that
two persons connected via a social relationship tend to have
similar tastes, which is known as “homophily principle” [3].
Social influence and social networks thus can help us predict a
target user’s preferences and decisions. In this section, we will
first briefly introduce a prevalent memory-based collaborative
filtering algorithm as a baseline algorithm, and then describe
our social influential recommendation model for individual
users in detail.
A. Collaborative Filtering
Collaborative filtering (CF) is one of the most successful
approaches to build a recommendation system. It uses the
known preferences of users to make recommendations or
predictions to the target user [1].
1) Similarity Computation: There are a variety of similarity measures. A generally adopted one is called Pearson
Correlation which measures the extent to which two variables
linearly relate with each other [2]. For user-based algorithm,
the Pearson Correlation between user u and v is
P
− r̄u )(rv,i − r̄v )
pP
,
2
2
i∈I (ru,i − r̄u )
i∈I (rv,i − r̄v )
wu,v = pP
i∈I (ru,i
(1)
where i ∈ I is the item rated by both users u and v, ru,i is
the rating of user u on item i, and r̄u is the average rating of
user u in co-rating set I.
2) Prediction Computation: Prediction Computation is the
most important step in a collaborative filtering system [1]. We
cf
can use the weighted sum to predict the rating Pu,i
for target
user u on a certain item i as proposed in [2]:
P
(rv,i − r̄v ) · wu,v
cf
Pu,i = r̄u + v∈UP
.
(2)
v∈U |wu,v |
Recommenders based on collaborative filtering then refer to
this prediction to provide top-k recommendations to the user or
simply display the personalized predicted rating of each item.
Unfortunately, they ignore the dependence and influence in
social networks in users preferences. Social Contagion Model
provides us an alternative solution to deal with this problem
by taking these effects into consideration.
fail to be activated to either state. Equation 4 states that
similar opinions to users’ expectations are easier and opposite
opinions are harder to be taken.
The state of newly activated node v under social influence
is decided by,


X
Sv,i = sign 
bvw · Sw,i 
(5)
B. Social Contagion Model
w∈N (v)
The flow of information through a social network can be
thought of as unfolding with the dynamics of an epidemic
[11]. This information from social relationships has potential
influence in people’s final buying decision. We use a linear
threshold social contagion model [10][11] to make recommendations with respect to this natural social contagion and user’s
subjective taste. We will start with a simple case that users use
“like” and “dislike” to describe their personal preferences and
explain the model in the perspective of game theory. We will
then discuss the multiple-scale rating case in Section III-B2.
1) A Simple Binary Rating Case: In a binary rating system,
we assume that users have three possible states namely like,
dislike and inactive. We use “1”, “-1” and “inactive-0” to label
them respectively. A node v is influenced by each neighbor
w according
to a weight/influence factor, bv,w ∈ [0, 1] such
P
that
w∈N (v) bv,w = 1. Intuitively, bv,w is related to trust
and interpersonal communication frequency. If we lack such
knowledge, we can randomly partition the unit influence
among neighbors of a user, which in fact simulates the
randomness of information cascade. The process proceeds as
follows: each node v chooses a threshold θv,i for item i from
the interval [0, 1], representing the weighted fraction of v’s
neighbors that must become active (either like or dislike in
order for v to become active to state like or dislike on item
i). Since θv,i indicates the latent tendency of nodes to adopt
the opinion, we naturally associate it with v’s CF prediction
on item i. E.g. we can set it as in Equation 4. Two other
classes of approaches are setting all thresholds uniformly at
random from the interval [0,1] or at a known value like 1/2.
Given a threshold, and an initial set of active nodes Ai (users
with non-empty ratings on a certain item i), this progressive
diffusion process proceeds deterministically.
Time operates in discrete steps t = 1, 2, 3, . . . . At a given
time t, any inactive node v becomes active if its fraction of
active neighbors exceeds its threshold:
X
bvw · Sw,i ≥ θv,i ,
(3)
w∈N (v)
where Sw,i is the state of node w on item i, and θv,i is the
influential threshold.
This in turn may cause other nodes to become active
in subsequent time steps, leading to potentially cascading
adoption behaviors of “like” or “dislike”. The process runs
until no more activations are possible.
For the target user u,
Su,i
if u is activated
si
Pu,i
=
.
(6)
inactive-0 otherwise
We note that if assuming the time interval during which a
user decides to buy the item recommended is short, then we
only need to consider the influence of the user’s immediate
friends under a unit time step or two steps, which means the
system only requires local social network information.
We can view this as a networked cooperation game [12].
Each node in the social network has three possible opinions
on item i: like, dislike, and inactive. Because of the homophily
effect of social networks, if two nodes are connected in the
social network, there is an incentive for them to have their
opinions match. We represent the payoff matrix as in Table I.
TABLE I: Payoff for the three-strategy coordination game
1
v -1
0
θv,i =
cf
Pv,i
,
if
cf
Pv,i
cf
min{Pv,i
,1
−
·
P
w∈N (v) bvw · Sw,i
cf
Pv,i }, otherwise
<0
. (4)
The intuition in Equations 3 and 4 is that if receiving too
many controversial opinions from neighbors, the node will
w
dislike(-1)
−avw , −awv
avw , awv
cf
0, −f (Pw,i
, −1)
inactive(0)
cf
−f (Pv,i
, 1), 0
cf
−f (Pv,i
, −1), 0
0, 0
Each node are playing many copies of this game with
all its neighbors at the same time. avw > 0 represents the
payoff/penalty node v receives if it coordinates/discoordinates
cf
with node w, and f (Pv,i
, ·) > 0 represents the penalty for
node v playing active strategy if its neighbor is inactive. The
total payoff is the sum of all payoffs of individual games. Then
the payoffs for node v are:
X
cf
• Payoff1 =
avw Sw,i − f (Pv,i
, 1)1{Sw,i =0}
w∈N (v)
•
Payoff−1 =
X cf
−avw Sw,i − f (Pv,i
, −1)1{Sw,i =0}
w∈N (v)
•
(
like(1)
avw , awv
−avw , −awv
cf
0, −f (Pw,i
, 1)
Payoff0 = 0
If node v chooses the strategy which provides maximum
cf
payoff and f (Pv,i
, ·) is reversely proportional to the number
of inactive nodes, the solution matches our social contagion
model.
(a) Threshold = 0.1.
(b) Threshold = 0.5.
(c) Threshold uniformly drawn from 0.05 to 0.8.
(d) Threshold = 0.1.
(e) Threshold = 0.5.
(f) Threshold uniformly drawn from 0.05 to 0.8.
Fig. 1: Simulations for binary rating social contagion process in a small-world social network of 1000 nodes.
2) General Case: For a 1−to−R scale rating system,
we pair up like’s and dislike’s in different levels. Denote R = {1, 2, · · · , R} as the set of ratings and S =
{±1, ±2, · · · , ±b R2 c} as the set of active state of different
levels of like’s and dislike’s. “active-0” ∈ S if R is odd. Define
the mapping function fr : R → S
fr (r) = r − r̄,
where
(7)
Here we assume that the mean-state “active-0” does not
provide valuable information thus it cannot influence other
nodes. We can easily change this assumption by making a
minor modification in Equation 8 and 9. The social influential
prediction on a target user u is
si
Pu,i
=
f −1 (Su,i ) if u is activated
.
inactive-0 otherwise
(10)
3) Simulation Results: We simulate the social contagion
process of binary rating model as shown in Fig.1. We construct
R
R
r̄ =
.
a social network generated by Watts and Strogatz model [13]
2
 R2
R
+
1
of 1000 nodes, and assign different thresholds for the nodes
2
2
E.g., for a 1-5 scale rating system, we assign “-2”, “-1”, and analyze the convergent rate and speed. Initially, active
“active-0”, “1”, “2” respectively on ratings 1 through 5. At nodes are in State like with probability 0.7 and dislike with
probability 0.3. We assign influential factors by a uniform
each time step, let
random partition of a unit influence among neighbors of a
X
node. Results are the average of 500 independent simulations.
S = arg max bvw · sign (Sw,i ) ,
(8) When the threshold is low, as in Fig.1a and Fig.1d, even a very
s∈S small portion of initial active nodes can activate almost all the
w∈N (v)
|S
w,i |=s
inactive nodes in the network, into both majority state like
and
minority state dislike. With the increasing number of the
P

inactive-0,
if
|
b
·
sign
(S
)
|
<
θ
initial
active nodes, the newly activated nodes dominatingly
w,i
v,i

w∈N (v) vw


|Sw,i |=S
!
choose the majority state and the number of iterations to
Sv,i =
.
P
convergence decreases. This shows that in a susceptible social


otherwise
w∈N (v) bvw · Sw,i ,
 S · sign
network, social contagion is likely to progressively influence
|Sw,i |=S
(9) all the nodes. When we increase threshold to 0.5, as shown in
θv,i in Equation 9 is the influential threshold.
Fig.1b and Fig.1e, social contagion process converges faster,
 R
 d2e
if R is odd
if R is even and r >
if R is even and r ≤
but small ratio of initial active nodes cannot influence the
network. Fig.1c and Fig.1f show a more realistic setting of
the threshold: each node’s threshold is drawn randomly from a
uniform distribution from 0.05 to 0.8, representing the various
susceptibility of different people. When 20% of nodes are
initially activated, more than half of the unactivated nodes will
become active to the majority states eventually, this rate stays
stable when the ratio of initial active nodes increases, while
the convergence speed increases. Our simulations show that
in a general setting, the result of social contagion process is
users’ local estimation of majority opinions with the penalty
from the disagreement with their own estimation. This also
follows our coordination game explanation. For more general
case, we expect that the ratings of newly activated nodes to be
clustered according to their social network structure, generally
following the majority opinion.
C. Discussions on Recommendation to Individuals
Define susceptibility factor αu ∈ [0, 1], which is the
attribute of a user. The more susceptible the user is, the higher
the value of αu should be. The recommendation prediction
Pu,i to a target user u on a certain item i is calculated by
(
cf
si
Pu,i
if Pu,i
= inactive-0
Pu,i =
cf
si
(1 − αu )Pu,i + αu Pu,i
otherwise
(11)
Then we can recommend the top-k predicted items to the
target user. In particular, if we set αu = 1 for ∀u, Equation
11 becomes
cf
si
Pu,i if Pu,i
= inactive-0
(12)
Pu,i =
si
Pu,i otherwise
If we ignore the inactive ratings, the recommendation model
above becomes a recommender based on ratings by users’
immediate and distant friends. It can be a supplement of
the existing recommendation systems and provide users extra
information of the opinions from someone they would trust.
IV. R ECOMMENDATIONS FOR G ROUPS
To show how our group recommendation model differs from
aggregating each user’s predictions or merging preference
profiles, let us begin with a brief example: Jessica and Mike
want to see a movie with Eric, a friend visiting them. When
choosing the movie to watch, the couple would like to follow
their guest’s opinion. Eric, accommodating as he is, also tries
to take Jessica and Mike’s taste into consideration. After some
discussion, they finally agree on a movie that Eric has wanted
to see for a long time, and also is interesting to both Jessica
and Mike. We will now introduce a social influence model to
describe Jessica, Mike and Eric’s opinion evolution in their
discussion.
formation, which postulates a recursive definition for the
interpersonal influence in a group of N users:
(t)
Pi
(t−1)
= AW Pi
(1)
+ (I − A)Pi ,
(13)
(1)
for t = 2, 3, . . .. Pi is an N × 1 vector of users’ initial
opinions on an item i, it can be either existing individual rating
(t)
ru,i or preference prediction Pu,i , or a mixture of both. Pi
is an N × 1 vector of users’ opinions at time t. W = [wuv ]
is an N ×PN matrix of interpersonal influences with 0 ≤
N
wuv ≤ 1, j wuv = 1, and A = diag(α11 , α22 , ..., αN N )
is an N × N diagonal matrix of users’ susceptibilities to
interpersonal influence on item i with 0 ≤ αuu ≤ 1. This
susceptibility factor is a user’s attribute and by default it is
associated with αu in Equation 11, but can also be set by
users. When A = 0, meaning no group members are willing
to change their own opinions in response to other group
(t)
(1)
members’ tastes. In that case, Pi = Pi . This is normally
what a traditional group recommender assumes. However,
although individuals in a group may not necessarily reach
consensus, they tend to coordinate their decisions through the
effect of social influence [20]. This is the initial motivation that
we apply social influence network theory to recommendation
model for groups.
When we consider the process in an equilibrium state
(assuming convergence), Equation 13 becomes
(∞)
Pi
(∞)
= AW Pi
(1)
+ (I − A)Pi .
(14)
If I − AW is nonsingular, then
(∞)
Pi
(1)
= V Pi ,
(15)
where
V = (I − AW )−1 (I − A).
(16)
V is a matrix describing the total interpersonal influence
(∞)
that affect a user’s evolution on a certain item. Pi
is an
N × 1 vector describing users’ final preference prediction on
a certain item i under social influence. This can be viewed as
a compromised disagreement after sufficient consideration on
N − 1 other group members’ opinions. Then we can follow
the normal regime, such as average rating, to aggregate users’
opinions to form a recommendation list for the group.
For example, in the scenario at the beginning of this section,
Jessica, Mike and Eric may set A = diag(0.9, 0.9, 0.5) and
W = [0.1 0.1 0.8; 0.1 0.1 0.8; 0.25 0.25 0.5]. For movie A,
(1)
suppose their individual predicted ratings are PA = (2, 3, 5).
(∞)
In above setting, by Equation 15 and 16, we have PA =
(4.52, 4.62, 4.86). Now the average rating of A for the group
(1)
is 4.66 instead of 3.33 if averaging directly from PA . We can
see that the result is inclined to Eric as it initially assumed.
A. Social Influence Model
B. Discussions on Possible Applications for Group Recommendation
Social influence theory [8][9] describes how a network
of interpersonal influence enters into the process of opinion
The recommendation model proposed in Section IV-A can
be applied and added to various recommendation systems with
different settings of interpersonal influence factor W and susceptibility factor A. For instance, for restaurant recommendation, it is likely that group members are willing to go to a good
restaurant even though some of them have already dined there
(1)
before. Thus original opinions Pi in Section IV-A can be a
combination of existing ratings and predicted ratings, and with
a higher susceptibility for the latter. On the other hand, for trip
recommendation and movie recommendation, group members
are likely to explore something unknown to everybody, thus
(1)
Pi are set as original predictions of individual preference.
In both cases, the recommendation model provides a way to
take interpersonal influence within a group into account before
preference aggregation procedure.
C. Social Contagion Model vs. Social Influence Model
Recommendation to individuals provides items that a target
user is most likely to be interested in. Social contagion model
in Section III is a probabilistic framework that simulates
how an opinion on a certain item spreads through the social
network. However, the buying decision made by the user is
a personal choice: the user may listen to others’ opinions but
there will be no compromise in the final decision. In addition,
the active nodes at the beginning are users who have already
made their explicit or implicit statement of preferences and
will not change with time. Since the time for the target user to
decide whether to buy the recommended item is usually short,
it is reasonable to model the social influence to individual
users as a progressive process. On the other hand, a single
decision for a group of people is not only based on fairness,
as in most of the current group recommenders, but also based
on group members’ susceptibility and influence power. We
assume group members will or are willing to modify their
choices according to other group members’ opinions, and it
is usually the case in social activities. Social influence model
shows us the compromised disagreement formed iteratively by
interpersonal influence.
V. C ONCLUSIONS AND FUTURE WORK
In this paper, we propose two social influential recommendation models, for individuals and for groups respectively.
The individual recommendation is based on social contagion,
which takes the effect of homophily among friends and
effect of word of mouth into consideration while making
recommendations. In the recommendation model for groups,
we provide preference specifications that not only reflect
subjective personal taste but also opinion evolution with the
knowledge of other members’ preferences.
For future work, we plan to validate our models through
a real online social network dataset from Yelp.com, which
provides users’ ratings of restaurants, spas, etc. Yelp also
provides social network feature which is ideal to compare
our model with the performance of collaborative filtering
recommendation and some existing recommendation systems
based on social networks such as SNRS [3]. We also want
to use machine learning techniques to cluster different types
(e.g. subjective, susceptible, hybrid, etc. ) of users, and explore
the well-performing settings for each type on threshold θu,i ,
susceptibility A and interpersonal influence weight W . Furthermore, we want to apply our group recommendation model
in different application areas such as movie recommendation,
restaurant application, etc.
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